In plain words
In that clarity all sorrows fall away, for the buddhi of one whose mind is serene quickly becomes steady.
Word for word
In serenity there arises for him the end of all sorrows; for the buddhi of one whose mind is serene quickly becomes firmly settled.
The commentators
Bhagavadgītābhāṣyam
प्रसादे सति किं स्यात् इत्युच्यते —
प्रसादे सर्वदुःखानाम् आध्यात्मिकादीनां हानिः विनाशः अस्य यतेः उपजायते । किञ्च — प्रसन्नचेतसः स्वस्थान्तःकरणस्य हि यस्मात् आशु शीघ्रं बुद्धिः पर्यवतिष्ठते आकाशमिव परि समन्तात् अवतिष्ठते, आत्मस्वरूपेणैव निश्चलीभवतीत्यर्थः ॥
एवं प्रसन्नचेतसः अवस्थितबुद्धेः कृतकृत्यता यतः, तस्मात् रागद्वेषवियुक्तैः इन्द्रियैः शास्त्राविरुद्धेषु अवर्जनीयेषु युक्तः समाचरेत् इति वाक्यार्थः ॥ ६५ ॥
In plain words
And when that serenity is actually there, what follows? To this the answer is given:
Once that clarity comes, every sorrow the striver carries — the ones that rise up in his own body and mind, and all the rest of them — is lost, and lost means destroyed. And there is more. For the one whose mind has gone clear, whose inner instrument now rests at home in itself, and precisely because of that, the intellect settles fast. Āśu: swiftly. Paryavatiṣṭhate — pari, all around, the way space is all around, and avatiṣṭhate, it stands and stays. It becomes unmoving as the very nature of the Self. That is what the line means.
So: for the one whose mind has cleared and whose understanding has come to rest, there is nothing left that has to be done. That is why, with the senses cut loose from liking and disliking, he stays collected and does what scripture does not forbid and what cannot be avoided. This is what the verse means.
Close to the Sanskrit
Serenity being present, what would be? — thus it is said:
When there is clarity (prasāda), the loss — that is, the destruction — of all sorrows, those beginning with the ones arising in the body and mind, comes about for this striver. Moreover: for one of clear mind, that is, for one whose inner instrument abides in itself, because of that the intellect quickly — swiftly — becomes fully established (paryavatiṣṭhate); like space, it abides pari, that is, on all sides. The meaning is that it becomes unwavering as the very nature of the Self.
Thus, since for one whose mind is clear and whose intellect is settled there is the state of having done all that is to be done — therefore, with senses unyoked from attraction and aversion, being composed, he should act in matters not contrary to scripture and unavoidable: this is the meaning of the sentence.